Tether paradoxes
Robert H. McNaught, Anglo-Australian Observatory (RMN@AAOCBN1.AAO.GOV.AU)
Sat, 16 Mar 1996 19:29:54 +1100 (EST)
Sue Worden wrote
>Question #1: How can a 0.1 inch diameter x 12.4 mile length (approx)
>cylinder appear to have the length-to-width ratio of a pencil?
>
>Question #2: How is it that we have been able to see something only
>0.1 inch wide at ranges in the hundreds of miles?
Answer Q 1 & 2
To understand these apparent paradoxes, it is necessary to differentiate
between apparent and actual angular diameter and the angular resolution
of the human eye.
It depends on what magnification you are using. The diameter of the tether
is not being resolved as it is subtends only 0.002 arcseconds at 300 km
distance. Thus, magnifying the tether in binoculars or a telescope will
not affect the angular diameter (until the seeing disk is resolved at the
arcsec level) whereas the length of the tether WILL be magnified.
At a suitably high magnification, when the seeing disk is just resolvable,
the apparent length-to-width ratio will be at its maximum and will be
considerably greater than the ratios of a pencil.
The resolution of the unaided eye is about 3 arcmin. Imagine two light
sources of the same total intensity, say Jupiter and a bright star. At the
naked-eye level, neither is resolved, both being smaller than 3' diameter,
but in a telescope, Jupiter IS resolved. So in other words, an object is seen
because it emits or reflects light, independent of whether it is of sufficient
angular diameter to be resolved. If the tether was say 60' long, then
the naked-eye length-to-width ratio of the tether would be 60:3 = 20:1.
In a telescope at 200 magnification, the apparent length is magnified to
60'x200=12,000' whereas, the apparent diameter is not magnified (being still
below the resolution). At this magnification, one can resolve a source
3'/200 ~= 1", so the length-to-width ratio would be 12,000:1.
Rob McNaught
rmn@aaocbn1.aao.gov.au